Dirac Convolutional Neural Networks for Simplicial Complexes

Higher-order systems capture interactions that go beyond the pairwise relationships represented by graphs and arise naturally in models of biological and physical systems. Simplicial complexes provide a common framework for representing such higher-order interactions, motivating the development of neural architectures that operate on them. In this work, we introduce Dirac Convolutional Networks (DCNs), a family of simplicial neural networks (SNNs) based on Dirac convolutions we introduce. Our construction is derived from a spectral filter formulation for multichannel, graded signals on simplicial complexes. We instantiate our framework with normalized root-to-leaf Dirac operator and propose Root-to-Leaf DCNs (R2L-DCNs) that inherit the benefits of normalized operators. First, normalization fixes the spectral scale of the Dirac operator and Hodge Laplacians, allowing filters to transfer across simplicial complexes. Second, it improves the stability of convolutional SNNs, allowing deep neural architectures. We establish this by proving that continuous, linear versions of these networks can preserve both the Dirichlet energy and the norm with right weights, and that their Euler discretizations are more stable when normalized operators are used. Finally, we empirically validate these properties across multiple benchmarks, showing that R2L-DCNs outperform existing SNN architectures on trajectory prediction and mesh classification benchmarks, and can maintain performance as network depth increases.